2015/08/04 by Charles Fefferman, Fefferman, Charles, Sergei Ivanov +7 · 2 citations
Mathematics · Computer Science · #Morphological variations and asymmetry #Numerical methods in inverse problems #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1508.00674
We study the geometric Whitney problem on how a Riemannian manifold (M,g)\ncan be constructed to approximate a metric space (X,dX). This problem is\nclosely related to manifold reconstruction where a smooth n-dimensional\nsubmanifold S\⊂ mathbb Rm, m>n needs to be constructed to\napproximate a point cloud in mathbb Rm. These questions are encountered\nin differential geometry, machine learning, and in many inverse problems\nencountered in applications. The determination of a Riemannian manifold\nincludes the construction of its topology, differentiable structure, and\nmetric.\n We give constructive solutions to the above problems. Moreover, we\ncharacterize the metric spaces that can be approximated, by Riemannian\nmanifolds with bounded geometry: We give sufficient conditions to ensure that a\nmetric space can be approximated, in the Gromov-Hausdorff or quasi-isometric\nsense, by a Riemannian manifold of a fixed dimension and with bounded diameter,\nsectional curvature, and injectivity radius. Also, we show that similar\nconditions, with modified values of parameters, are necessary.\n As an application of the main results we give a new characterisation of\nAlexandrov spaces with two-sided curvature bounds. Moreover, we characterise\nthe subsets of Euclidean spaces that can be approximated in the Hausdorff\nmetric by submanifolds of a fixed dimension and with bounded principal\ncurvatures and normal injectivity radius.\n We develop algorithmic procedures that solve the geometric Whitney problem\nfor a metric space and the manifold reconstruction problem in Euclidean space,\nand estimate the computational complexity of these procedures.\n